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| Mirrors > Home > MPE Home > Th. List > cnconst | Structured version Visualization version GIF version | ||
| Description: A constant function is continuous. (Contributed by FL, 15-Jan-2007.) (Proof shortened by Mario Carneiro, 19-Mar-2015.) |
| Ref | Expression |
|---|---|
| cnconst | β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ (π΅ β π β§ πΉ:πβΆ{π΅})) β πΉ β (π½ Cn πΎ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fconst2g 7221 | . . . 4 β’ (π΅ β π β (πΉ:πβΆ{π΅} β πΉ = (π Γ {π΅}))) | |
| 2 | 1 | adantl 480 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ:πβΆ{π΅} β πΉ = (π Γ {π΅}))) |
| 3 | cnconst2 23207 | . . . . 5 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ) β§ π΅ β π) β (π Γ {π΅}) β (π½ Cn πΎ)) | |
| 4 | 3 | 3expa 1115 | . . . 4 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (π Γ {π΅}) β (π½ Cn πΎ)) |
| 5 | eleq1 2817 | . . . 4 β’ (πΉ = (π Γ {π΅}) β (πΉ β (π½ Cn πΎ) β (π Γ {π΅}) β (π½ Cn πΎ))) | |
| 6 | 4, 5 | syl5ibrcom 246 | . . 3 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ = (π Γ {π΅}) β πΉ β (π½ Cn πΎ))) |
| 7 | 2, 6 | sylbid 239 | . 2 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ π΅ β π) β (πΉ:πβΆ{π΅} β πΉ β (π½ Cn πΎ))) |
| 8 | 7 | impr 453 | 1 β’ (((π½ β (TopOnβπ) β§ πΎ β (TopOnβπ)) β§ (π΅ β π β§ πΉ:πβΆ{π΅})) β πΉ β (π½ Cn πΎ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 {csn 4632 Γ cxp 5680 βΆwf 6549 βcfv 6553 (class class class)co 7426 TopOnctopon 22832 Cn ccn 23148 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-topgen 17432 df-top 22816 df-topon 22833 df-cn 23151 df-cnp 23152 |
| This theorem is referenced by: xrge0mulc1cn 33575 cxpcncf2 45316 |
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